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Chapter 10: Problem 3

Graph the plane curve given by the parametric equations. Then find an equivalent rectangular equation. $$x=4 t^{2}, y=2 t ;-1 \leq t \leq 1$$

Short Answer

Expert verified
Plot the parametric curve points, then convert to rectangular form: \[ x = y^2 \]

Step by step solution

01

- Understand the Parametric Equations

The given parametric equations are: \[ x = 4t^2 \] \[ y = 2t \] with the parameter range: \[ -1 \leq t \leq 1. \]
02

- Calculate the Bounds for x and y

Calculate the bounds for the parameter range: For \[ t = -1 \] \[ t = 0 \] \[ t = 1 \]When \[ t = -1 \] \[ x = 4\] \[ y = -2 \]When \[ t = 0 \] \[ x = 0 \] \[ y = 0 \]When \[ t = 1 \] \[ x = 4 \] \[ y = 2 \]
03

- Plot Key Points from the Parametric Equations

Plot the points on a graph obtained from different parametric values: \[ (-2, 4), (0, 0), (2, 4) \] and connect them to form the shape of the curve.
04

- Derive the Rectangular Equation

Eliminate the parameter \( t \ \text{from the equations}\): Solve \( y = 2t \) for \( t \): \[ t = \frac{y}{2} \]Substitute this into \( x = 4t^2 \): \[ x = 4 \left(\frac{y}{2}\right)^2 \] which simplifies to: \[ x = y^2 \]
05

- State the Rectangular Equation

The equivalent rectangular equation of the curve is: \[ x = y^2 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Parametric Equations
Parametric equations define a set of related values, often represented as functions of an arbitrary parameter, usually denoted as 't'. In our example, the given parametric equations are:
  • \(x = 4t^2\)
  • \(y = 2t\)
Here, 't' can range from -1 to 1. These equations generate pairs of (x, y) values that can be plotted on a graph. To graph these points, select specific values for the parameter 't' within its range, compute the corresponding 'x' and 'y', and plot the points.When \(t = -1\):
  • x = 4(-1)^2 = 4
  • y = 2(-1) = -2

When \(t = 0\):
  • x = 4(0)^2 = 0
  • y = 2(0) = 0

When \(t = 1\):
  • x = 4(1)^2 = 4
  • y = 2(1) = 2

By plotting these points:
  • (4, -2)
  • (0, 0)
  • (4, 2)
and smoothly connecting them, you form a curve representing the parametric equations.
Rectangular Equations
To understand rectangular equations, remember they are the standard form of equations, typically involving only 'x' and 'y'. When dealing with parametric equations, it's often helpful to convert them to rectangular form to simplify graphing or analysis. Rectangular equations eliminate the parameter, directly relating x and y instead.In our example, the parametric equations \(x = 4t^2\) and \(y = 2t\) can be converted to a single rectangular equation. Identifying the relationship between \(x\) and \(y\) helps to find:
  • a more conventional representation of the curve
  • a clearer understanding of the shape and properties of the graph

The parametric form often benefits from graphing to see the connection between x and y indirectly through the parameter.
Eliminating the Parameter
To convert the parametric equations to a rectangular form, eliminating the parameter is key. This process involves expressing 't' in one equation and substituting it into the other.For our given equations:
  • Solve \(y = 2t\) for 't': \ t = \dfrac{y}{2} \.
  • Substitute this value into the equation of 'x': \ x = 4\bigg( \dfrac{y}{2} \bigg)^2 \
  • Simplify: \ x = \dfrac{4y^2}{4} = y^2 \.
Thus, the rectangular equation is \(x = y^2\), which directly relates 'x' and 'y', eliminating the parameter 't.'This equivalent rectangular form is easier for most to visualize, as it clearly shows that 'x' is always non-negative (since it鈥檚 the square of 'y') and accurately represents the symmetry of the original parametric equations.

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Most popular questions from this chapter

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